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Chapter 10: Problem 3

To test \(H_{0}: \sigma=50\) versus \(H_{1}: \sigma<50,\) a random sample of size \(n=24\) is obtained from a population that is known to be normally distributed. (a) If the sample standard deviation is determined to be \(s=47.2,\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.05\) level of significance, determine the critical value. (c) Draw a chi-square distribution and depict the critical region. (d) Will the researcher reject the null hypothesis? Why?

Short Answer

Expert verified
Test statistic: 20.4957; Critical value: 13.091; Do not reject the null hypothesis.

Step by step solution

01

Define Test Statistic

First, calculate the test statistic for the standard deviation. For a sample of size n, the test statistic for variance is given by \[\chi^2 = \frac{(n-1)s^2}{\sigma_0^2}.\]Here, n is the sample size, s is the sample standard deviation, and \(\sigma_0\) is the hypothesized population standard deviation.
02

Substitute the Values

Plug in the given values into the formula:\[\chi^2 = \frac{(24-1) \times (47.2)^2}{(50)^2}.\]Calculate this step by step:n = 24,\quad s = 47.2,\quad \sigma_0 = 50Then,\[\chi^2 = \frac{23 \times 47.2^2}{50^2}.\]
03

Compute the Test Statistic

Simplify the expression from step 2:\[\chi^2 = \frac{23 \times 2227.84}{2500}.\]Calculate:\[23 \times 2227.84 = 51239.32 \chi^2 = \frac{51239.32}{2500} = 20.4957.\]
04

Determine the Critical Value

For a given level of significance, \(\alpha = 0.05\), and n-1 degrees of freedom (df = 23), we look up the critical value in the chi-square distribution table for a one-tailed test. The critical value for \(\chi^2_{0.05, 23}\) is approximately 13.091.
05

Draw the Chi-Square Distribution

Draw a chi-square distribution curve. Mark the critical value (13.091) on the x-axis and shade the region to the left of this value. This is the critical region.
06

Decision

Compare the test statistic (20.4957) with the critical value (13.091). Since 20.4957 is greater than 13.091, it falls outside the critical region. Hence, do not reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

test statistic
The test statistic is a crucial part of hypothesis testing. It assists in determining whether the null hypothesis can be rejected. In the context of the chi-square test for standard deviation, the test statistic is calculated using the formula: \(\backslash chi^2 = \frac{(n-1)s^2}{\backslash sigma_0^2}\).

Here's what each term represents:

  • \(n\): Sample Size
  • \(s\): Sample Standard Deviation
  • \(\sigma_0\): Hypothesized Population Standard Deviation


By using the given values: \(n = 24\), \( s = 47.2\), and \( \sigma_0 = 50\), we substitute them into the formula:

  • \(\backslash chi^2 = \frac{23 \times 47.2^2}{50^2}\)


This simplification renders:
  • \(23 \times 47.2^2 = 51239.32\)
  • \( \frac{51239.32}{2500} = 20.4957 \)

Therefore, the test statistic is 20.4957.
critical value
The critical value is a threshold which the test statistic is compared against to decide whether to reject the null hypothesis. For a chi-square test with a given significance level (\(\alpha\)) and degrees of freedom (df), we reference the chi-square distribution table to find this value.

For our example, we have a significance level of \(\alpha = 0.05\) and degrees of freedom \( (n-1) = 23 \). Consulting the chi-square distribution table, the critical value for a one-tailed test is approximately 13.091. This value divides the chi-square distribution into two regions:
  • The region where the null hypothesis is not rejected (to the right of the critical value).
  • The critical region where the null hypothesis is rejected (to the left of the critical value).

The critical value acts as a benchmark to assess the test statistic.
rejection region
The rejection region is a part of the chi-square distribution where, if the test statistic falls within, the null hypothesis is rejected. It is defined by the critical value.

In our example, the critical value at \(\alpha = 0.05\) with 23 degrees of freedom is 13.091. This means:
  • If the test statistic \( \chi^2 \) is less than 13.091, we reject the null hypothesis.
  • If the test statistic is greater than or equal to 13.091, we do not reject the null hypothesis.


The rejection region is typically shaded on the chi-square distribution curve, to the left of the critical value. Since our test statistic (20.4957) is greater than the critical value, it falls outside the rejection region. Hence, we do not reject the null hypothesis.
Understanding the rejection region is essential as it visually represents the decision rule in hypothesis testing.

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Most popular questions from this chapter

Explain what it means to make a Type II error.

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